Finite Group Databasebeta

← #11720999

Subgroups — #11720999

Order 864.

248 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×D9C211yesmaximal, characteristic
2C2×#6189C211yesmaximal, characteristic, complemented
3#88534C211yesmaximal, characteristic, complemented
4C3×(C2×D9)⋊C4C211yesmaximal, characteristic
5C9×C4⋊Dic3C211yesmaximal, characteristic, complemented
6(C2×D9)⋊Dic3C211yesmaximal, characteristic
7#88442C211yesmaximal, characteristic, complemented
8#63899—31nomaximal
9#63458—31nomaximal
10C22×C3×D9C2211yescharacteristic
11C2×C3×C4×C9C2211yescharacteristic
12C2×C3⋊Dic9C2211yescharacteristic
13C2×C3⋊Dic9C2211yescharacteristic
14C2×C9×Dic3C2211yescharacteristic
15C2×C9×Dic3C2211yescharacteristic
16C2×C3×Dic9C2211yescharacteristic
17Dic3×D9—22no—
18#6189—22no—
19C2×#627—31no—
20C2×D3×Dic3—31no—
21#3750—31no—
22C3×(C2×D3)⋊C4—31no—
23C3×C4⋊Dic3D311yescharacteristic
24#3724—31no—
25(C2×D3)⋊Dic3—31no—
26C2×C4×D9—31no—
27C2×Dic18—31no—
28C9×C4⋊C4—31no—
29(C2×D9)⋊C4—31no—
30(C2×D9)⋊C4D311yescharacteristic
31Dic9⋊C4—31no—
32Dic9⋊C4—31no—
33Dic9⋊C4—61no—
34C22×C3×C9C2311yescharacteristic
35C2×C3×D9—21no—
36C2×C3×D9—21no—
37C2×C3×D9Q812yescomplemented
38C3×C4×C9—21no—
39C3⋊Dic9—21no—
40C3⋊Dic9D412yes—
41C9×Dic3—21no—
42C9×Dic3—21no—
43C3×Dic9—21no—
44#896—91no—
45#878—91no—
46C22×C3×D3—31no—
47C2×C32×C4C2×D311yescharacteristic
48C2×C3⋊Dic3—31no—
49C2×C3⋊Dic3—31no—
50C2×C3×Dic3—31no—
51C2×C3×Dic3C2×D311yescharacteristic
52C2×C3×Dic3C2×D311yescharacteristic
53#627—62no—
54D3×Dic3—62no—
55C22×D9C2×D311yescharacteristic
56C2×C4×C9—21no—
57C2×C4×C9—31no—
58C2×C4×C9—31no—
59C2×C4×C9C2×D311yescharacteristic
60C2×Dic9—31no—
61C2×Dic9—31no—
62C2×Dic9—61no—
63C2×Dic9—61no—
64C2×Dic9C2×D311yescharacteristic
65C4×D9—62no—
66Dic18—62no—
67C2×C3×C9C2×D411yescharacteristic
68C2×C3×C9C2×Q811yescharacteristic
69C2×C3×C9#4111yescharacteristic
70C3×D9—22no—
71C22×Dic3—91no—
72C2×C4×D3—91no—
73C2×Dic6—91no—
74C2×Dic6—91no—
75C3×C4⋊C4—31no—
76C3×C22⋊C4—91no—
77#217—91no—
78(C2×D3)⋊C4—31no—
79(C2×D3)⋊C4—91no—
80C4⋊Dic3D911yescharacteristic, complemented
81Dic3⋊C4—91no—
82Dic3⋊C4—91no—
83Dic3⋊C4—91no—
84Dic3⋊C4—91no—
85Dic3⋊C4—181no—
86C22×C32C22×D311yescharacteristic
87C2×C3×D3—32no—
88C2×C3×D3—61no—
89C2×C3×D3—61no—
90C32×C4—21no—
91C3⋊Dic3—32no—
92C3⋊Dic3—61no—
93C3×Dic3—21no—
94C3×Dic3—21no—
95C3×Dic3—61no—
96C22×C9—21no—
97C22×C9C22×D311yescharacteristic
98C2×D9—21no—
99C2×D9—21no—
100C2×D9Dic612yescomplemented
101C4×C9—21no—
102C4×C9—41no—
103C4×C9—61no—
104C4×C9—61no—
105Dic9—21no—
106Dic9—32no—
107Dic9—62no—
108Dic9—61no—
109Dic9—121no—
110C22⋊Q8—271noSylow
111C3×C9C22⋊Q811yescharacteristic, complemented, Sylow
112C23×C3—91no—
113C22×D3—31no—
114C2×C3×C4—21no—
115C2×C3×C4—31no—
116C2×C3×C4—31no—
117C2×C3×C4—91no—
118C2×C3×C4C2×D911yescharacteristic
119C2×C3×C4D3211yescharacteristic
120C2×Dic3—31no—
121C2×Dic3—91no—
122C2×Dic3—91no—
123C2×Dic3—91no—
124C2×Dic3—91no—
125C2×Dic3—181no—
126C2×Dic3—181no—
127C2×Dic3—182no—
128C2×Dic3C2×D911yescharacteristic
129C2×Dic3C2×D911yescharacteristic
130C4×D3—182no—
131Dic6—182no—
132Dic6—182no—
133C2×C32#23511yescharacteristic
134C2×C32D3×D411yescharacteristic
135C2×C32D3×Q811yescharacteristic
136C3×D3—62no—
137C2×C9—21no—
138C2×C9—21no—
139C2×C9—21no—
140C2×C9C2×Dic611yescharacteristic
141C2×C9D3×D411yescharacteristic
142C2×C9#23711yescharacteristic
143D9—22no—
144C2×Q8—271no—
145C22×C4—271no—
146C4⋊C4—31no—
147C4⋊C4—271no—
148C4⋊C4—271no—
149C22⋊C4—91no—
150C22⋊C4—271no—
151C22×C3—21no—
152C22×C3—92no—
153C22×C3—181no—
154C22×C3—181no—
155C22×C3C22×D911yescharacteristic
156C22×C3C2×D3211yescharacteristic
157C2×D3—32no—
158C2×D3—61no—
159C2×D3—61no—
160C3×C4—21no—
161C3×C4—21no—
162C3×C4—41no—
163C3×C4—61no—
164C3×C4—61no—
165C3×C4—181no—
166Dic3—21no—
167Dic3—21no—
168Dic3—61no—
169Dic3—92no—
170Dic3—92no—
171Dic3—182no—
172Dic3—181no—
173Dic3—181no—
174Dic3—361no—
175C32#89611yescharacteristic
176C9—21no—
177C9#87811yescharacteristic, complemented
178C23—91no—
179Q8—542no—
180C2×C4—31no—
181C2×C4—31no—
182C2×C4—91no—
183C2×C4—271no—
184C2×C4—271no—
185C2×C4—542no—
186C2×C4D3×D911yescharacteristic
187C2×C3—21no—
188C2×C3—21no—
189C2×C3—21no—
190C2×C3—182no—
191C2×C3#369711yescharacteristic
192C2×C3D4×D911yescharacteristic
193C2×C3Q8×D911yescharacteristic
194C2×C3D3×Dic611yescharacteristic
195C2×C3#380211yescharacteristic
196C2×C3#380411yescharacteristic
197D3—62no—
198C22—92no—
199C22—181no—
200C22—181no—
201C22C2×D3×D911yescharacteristic
202C4—21no—
203C4—61no—
204C4—61no—
205C4—181no—
206C4—272no—
207C4—541no—
208C3—21no—
209C3#6345811yescharacteristic, complemented
210C3#6389911yescharacteristic
211C2—182no—
212C2D9×Dic611yescharacteristic
213C2#8864611yescharacteristic
214C2#8865811yescharacteristic
subgroupG/H
centerC22C2×D3×D9
commutator, cosocleC22×C3×C9C23
FrattiniC22×C3C2×D32
FittingC2×C3×C4×C9C22
socleC22×C32C22×D3
prime radicalC22×C3×D9C22
hypercenterC2×C4D3×D9
hypocenterC3×C9C22⋊Q8
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#6189C2semidirect
3. #88534C2semidirect
5. C9×C4⋊Dic3C2semidirect
7. #88442C2semidirect
37. C2×C3×D9Q8semidirect
80. C4⋊Dic3D9semidirect
100. C2×D9Dic6semidirect
111. C3×C9C22⋊Q8semidirect
177. C9#878semidirect
209. C3#63458semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×Dic3×D92
C2×#61892
#885342
C3×(C2×D9)⋊C42
C9×C4⋊Dic32
(C2×D9)⋊Dic32
#884422
#638993
#634583